1. Find following intergrals. a) ∫ sin(2x)cos(3x) dx b) ∫ (x-1)^2 * e^(2x) dx c) ∫ e^(2x+1) * sin(πx/2) dx d) ∫ [ x^2 / (4x^2 + 4x + 1)] dx 2. Find a solution of dy/dx = y^2 - 4 that passes through (0,4) 3. Let L(t) be the length of a fish at time t. Suppose that the fish grows according to the von Bertalanffy equaton dL/dt = k(34 - L) with L(0)=2 a) Solve the differential equation b) Determine k under the assumption that L(4)=10 c) Find the length of the fish when t=10 d) Find lim(t→∞)L(t) 4. a) Find the Taylor polynomial for f(x) = arctan x about x=0 for │x│≦1 b) Explain why the following holds: (π/4) = 1 - (1/3) + (1/5) - (1/7) + ... 5. Discuss the convergence and divergence of ∫(0到∞) (x^p) * (e^(-x)) dx for 0＜p＜∞ 6. Determine and explain whether integral is convergent a) ∫(1到∞) [1/√(1 + 6^x)] dx b) ∫(1到∞) [1/√(x + ln x)] dx 7. a) How large should n be so that the midpoint rule approximation of ∫(0到2) x^2 dx is accurate to within 10^(-4) b) How large should n be so that the trapezoidal rule approximation of ∫(0到2) x^2 dx is accurate to within 10^(-4) --

※ 发信站: 批踢踢实业坊(ptt.cc) ◆ From: 140.119.180.22